Journal of Theoretical Medicine, 2002 Vol. 4 (2), pp. 95–108
Boundary-layer Type Solutions for Initial Platelet
Activation and Deposition
T. DAVIDa,*, P.G. DE GROOTb and P.G. WALKERa
a
Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand; bDepartment of Haematology,
University Hospital Utrecht, Utrecht, The Netherlands
(Received 7 April 2000; Revised 22 January 2002; In final form 25 April 2002)
This paper presents, on the basis of high Peclet number, a mathematical model for the activation and
initial adhesion of flowing platelets onto a surface. In contrast to past work, the model is applicable to
general 2D and axi-symmetric flows where the wall shear stress is known a priori. Results indicate that
for high activation reaction rates there exist two layers, one containing only activated platelets and the
other both activated and non-activated platelets. Fundamental relationships are proposed between the
adhesion rate of platelets to the surface and the characteristic parameters of Peclet number and
Reynolds number. Activation in the bulk fluid (blood) is characterised by the Damkohler number,
which is a function of activation rate and the free-stream velocity. It is shown that, as the free-stream
velocity varies, there exists a maximum of activated platelet flux to the wall for particular values of the
velocity. These values, at which the maximum occur, are themselves functions of the platelet activation
rate. As the free-stream velocity increases the activation of platelets ceases altogether and adhesion is
reduced to a very small value strengthening the hypothesis of the correlation between
atherogenesis/thrombogenesis and areas of low shear.
Keywords: Platelet adhesion; Boundary layer flow
INTRODUCTION
Atherosclerosis is one of the leading causes of death in
the world today. In the United States alone, in 1995,
there were nearly half a million deaths attributable to
coronary disease (Wilson and Ferguson, 1999). It is now
reasonably well established that there exists a strong
relationship between the flow of blood, the permeability
of the endothelium to cholesterol and other LDLs (lowdensity lipoproteins) leading to the formation of
atherosclerotic plaques. Platelet adhesion and subsequent aggregation are also important in atherogenesis,
along with endothelial dysfunction and atherosclerotic
plaque instability. In addition, the first stage of
thrombogenesis is platelet adhesion on a surface
followed by aggregation and the formation of platelet
mural thrombi (Friedman and Leonard, 1971). The
resulting thrombosis and/or embolisation from diseased
*Corresponding author. E-mail: t.david@mech.canterbury.ac.nz
ISSN 1027-3662 print/ISSN 1607-8578 online q 2002 Taylor & Francis Ltd
DOI: 10.1080/1027366021000003261
arteries produces a wide variety of clinical scenarios;
myocardial infarction, strokes and gangrene. Vascular
geometry can now be regarded as a risk factor due to
the influence of that geometry on local haemodynamic
effects. In particular, it is extremely important to gain a
thorough understanding of the relationship between
vascular geometry, blood flow and the onset of
thrombus formation in both the natural and diseased
arteries. Although the formation of thrombi and platelet
activation in stasis is fairly well understood, the
influence of blood-flow characteristics has yet to be
fully investigated. Fluid dynamic studies of blood flow,
in models of arteries, suggest a set of fluid dynamic
conditions that appear to predispose thrombus formation
(platelet adhesion), principally at arterial bifurcations,
T-junctions and curved sections.
Over the past 20 years, there have been a number of
proposed models for platelet adhesion and its relationship
96
T. DAVID et al.
to blood flow and vascular geometry (Leonard et al., 1972;
Turitto and Baumgartner, 1975; Kratzer and Kinder, 1986;
Affeld et al., 1995) as well as a considerable amount of
experimental data collected (Petschek et al., 1968;
Friedman and Leonard, 1971; Leonard et al., 1972;
Turitto and Baumgartner, 1975; Kratzer and Kinder,
1986; Strong et al., 1987; Tippe et al., 1992; Reininger
et al., 1993; Schoephoerster et al., 1993; Affeld et al.,
1995; Reininger et al., 1996). The models concentrated on
both stagnation point flow as well as simple linear shear
flow. However, initially it was assumed that a diffusioncontrolled system existed where the adhesion rate was
large compared to the diffusion coefficient. Recently,
workers have begun to call into question the diffusioncontrolled system and look more closely at intermediate
kinetics. Wall shear stress has been proposed as a possible
mediator for platelet adhesion (Rajagopalan et al., 1988;
Weiss, 1995).
Experiments have been done in parallel plate
flow chambers, annular expansion tubes (Turitto and
Baumgartner, 1975; Karino and Goldsmith, 1979) and in
stagnation flow regimes. However, the main theoretical
analysis has been attempted (Turitto and Baumgartner,
1975; Strong et al., 1987) only for the parallel flow
system, where wall shear rate is constant. Numerical
models have been developed that include the integration
of computational fluid dynamics with relatively complex
kinetic mechanisms (Sorensen et al., 1999a,b). However,
the flow regime was of constant shear type and the
presented results concentrated on obtaining reaction-rate
parameters from comparison with experiment, although
this model could very well be used for more complex flow
conditions. Affeld et al. (1995) investigated platelet
adhesion in a stagnation point flow chamber. The results
from the experiment indicated, as had others, that the
small neighbourhood of the stagnation point streamline
was devoid of platelets along with the existence of a
domain of maximum platelet adhesion just downstream of
the stagnation point streamline. They put forward the
hypothesis that thrombin emanating from the adhered
platelet granules would be convected downstream and
activate other platelets flowing above the adhesion surface
and that the maximum adhesion rate occurred for a
“critical wall shear rate”.
The presented work puts forward a boundary-layer type
model for platelet activation and adhesion. This model is
applicable to axi-symmetric flows, commonly found in the
neighbourhoods of stagnation points and 2D flows where,
in both cases, the wall shear stress is known a priori. It
covers a large range of kinetic rates, from reaction
controlled through intermediate kinetics to diffusioncontrolled systems. We investigate the role of thrombin,
emanating from adhered platelets on the activation of
flowing non-activated platelets and we compare the results
with experiment. The model provides a parametric
representation of platelet activation and adhesion through
the three fundamental parameters of Peclet, Reynolds
and Damkohler numbers and indicates a fundamental
relationship between adhesion rate and the roles of
convection and diffusion.
MATHEMATICAL MODEL
In order to fully understand the mechanism of thrombogenesis, it is important to investigate the chemicals
important to platelet activation. One proposed mechanism
(Hubbell and McIntire, 1986) is that adherent platelets, if
sufficiently activated, produce local high concentrations of
platelet activating substances, which then diffuse and
convect into the flow thereby activating platelets in the bulk
flow. These highly reactant chemicals are important in
determining the complex chemical mechanisms, which
make up the clotting cascade; they all essentially diffuse
outward from the activated and adhered platelet. This may
then activate other platelets that may be convected and/or
diffused into the reaction neighbourhood.
Three chemicals have been identified as playing
important roles in platelet activation;
Adenosine diphosphate: released from the dense
granules of the platelet.
Thromboxane A2: enzymatically generated on or near
the membrane of the platelet.
Thrombin: secreted from platelets both when activated
and adhering.
From a modelling viewpoint, we may simplify this
system somewhat by lumping these chemicals into a
single species, which for the purpose of clarity, shall be
denoted as thrombin. Both activated and non-activated
platelets can adhere to the wall or surface although this
will be at different rates.
Basic Theory
In this section, we set out the basic conservation equations
for both fluid flow and reactive species. In order to compare
with other workers, we assume that firstly blood be
modelled as a Newtonian fluid, secondly that the system is
of constant density and temperature and finally, although
physiological blood flows are pulsatile in nature, we
assume a steady-state system. The steady-state conservation of mass and momentum in vector form is written as
7·u ¼ 0
ð1Þ
2
u
r 7
2 u £ ð7 £ uÞ ¼ 27p þ m72 u
2
ð2Þ
and
Here u is the velocity vector in a general orthogonal set of
co-ordinates, r is the density, m is the dynamic viscosity
and p is the pressure that varies due to dynamic variations
PLATELET ACTIVATION AND DEPOSITION
in fluid velocity alone. Red blood cells (RBC) have a
considerable effect on the diffusion of platelets, an effect
that is enhanced by shear rate. However, the main
experimental evidence with which we compare our
analysis was done with platelet-rich plasma containing no
RBCs (Affeld et al., 1995). Hence for the present we
assume constant diffusion coefficients.
Although adhesion is a time-dependent process for
initial adhesion Strong et al. (1987) have shown that for
small times adhesion is the product of a constant rate of
adhesion and time. We can assume steady state and then
use a simple product of platelet flux and small time to
evaluate the number of platelets adhering. Hence, for this
present model, we look at steady state only (the similarity
solution method is not restricted to steady-state methods
see David et al., 2001a). A general conservation equation
for the ith cellular/chemical species can be written, again
in vector form, as
7·ðufi Þ ¼ 7·ðDi 7fi Þ þ W i
i ¼ 1; . . .; N
ð3Þ
fi is the ith mass fraction defined as fi ¼ ri =r; Wi is the
rate of production of the ith species mass fraction and Di is
the ith species diffusion coefficient assuming a Fickian
diffusion model. Since the bulk fluid density is constant,
the momentum and species-conservation equations are
effectively de-coupled.
The diffusion coefficient for platelets may be
determined by the use of the Stokes –Einstein equation
Dpl ¼
BT
6pmr pl
ð4Þ
Here B is the Boltzmann constant, T is the absolute
temperature, m is the coefficient of viscosity of plasma and
rpl is the radius of a platelet. Using the above and an
assumption of a core-body temperature of 378 for T, Dpl,
the platelet diffusion coefficient was calculated as
1.7 £ 10213 m2 s21. This compares favourably with
values calculated by many other workers (Strong et al.,
1987). If RBC augmentation is modelled, then this
diffusion coefficient value can increase by as much as
two orders of magnitude (Strong et al., 1987), depending on
the shear rate.
For the case presented here, it was assumed that platelet
activation occurred in the bulk fluid, through a reaction
with thrombin, and that the phenomenon of platelet
adhesion to the wall is represented by a simple reaction
boundary condition, where platelets are either “free” or
permanently adhered. Once adhered, it is assumed that
thrombin is diffused outward from the adhered platelet at a
specified rate. The thrombin emanates from the internal
part of the cell and it has been shown that the time taken to
diffuse out is quite long due to the canalicular structure of
the platelet membrane (Fogelson and Wang, 1996). For
this model, we can assume that although the platelet may
be activated in the bulk fluid, it does not release thrombin
until adhered at the surface. In addition, it is assumed that
97
no reversible reactions occur representing the worst-case
scenario for thrombogenesis. The reaction mechanisms
are given by
kb
thrombin þ platelets ! activated platelets þ thrombin
kwa
surface þ activated platelets ! adhered platelets þ surface
ð5Þ
kwp
surface þ platelets ! adhered platelets þ surface
kwt
adhered platelets ! adhered platelets þ thrombin
Here kb, is the forward rate for the bulk reaction and kwa,
kwp and kwt are surface (or wall) reactions for activated
platelets, non-activated platelets and thrombin, respectively. The constraint that thrombin is effused from an
activated platelet only when it has adhered can be relaxed
so that the first reaction mechanism given in Eq. (5) can be
rewritten as
kb
thrombin þ platelets ! activated platelets
ð6Þ
þ a thrombin
where a . 1; this essentially corresponds to a non-zero
source term in the conservation equation for thrombin. We
assume that the both platelets and activated platelets
become adhered at the surface at a rate proportional to
their concentration at the wall in a similar manner to that
given by Sorensen et al. (1999b). We can write these
surface boundary conditions as
Dact pl
›fact pl ¼ kwa fact pl jsurface
›n surface
›fpl Dpl
¼ kwp fpl jsurface
›n surface
ð7Þ
Here n is the co-ordinate normal to the reacting surface.
Finally, the production of thrombin is determined by the
rate of adhesion of platelets. So that in a similar manner
we can write
›fth ¼ 2kwt ½kwa fact pl þ kwp fpl Š
Dth
›n surface
ð8Þ
The modelling procedure has been split into two distinct
areas, an analytical solution for the limiting case of infinite
Damkohler number and a numerical procedure for the
general coupled equations of species conservation.
The analysis is applicable to both axi-symmetric and 2D
flows and Fig. 1(a) and (b) shows the co-ordinate systems
for both cases
For clarity the fluid domain will be assumed to be of an
axi-symmetric form, modelling the stagnation point flow.
However, this analysis may be used for any a priori known
98
T. DAVID et al.
FIGURE 1 (a) Co-ordinate system for stagnation point flow. (b) Co-ordinate system for parallel plate flow.
2D velocity field where the boundary-layer flow does not
detach. We concentrate on phenomena in the viscous
boundary layer and the mass-transfer sub-layer and as
such use a boundary-layer formulation. As a first step in
the solution process, we assume that mass diffusion in the
stream-wise radial (R̃ ) direction is negligible compared to
that in the axial (z̃ ) direction. With these assumptions, the
constant density conservation equation for the mass
fraction of the ith cellular species, fi, can be written in the
boundary-layer type form (Schlichting, 1960) as
u~
2 ~
›f~i
›f~i
~ i › fi þ W
~i
¼D
þ v~
›z~
›z~ 2
›R~
i ¼ 1; . . .; N
ð9Þ
where u~ and v~ are the velocity components for the R~ and z~
directions, respectively, and N is the total number of
species participating within the domain. Again, in a
similar manner to Sorensen et al. (1999b), Wi can be
modelled as a first-order reaction mechanism so that
~ i ¼ kib ½fk Š½fl Š
W
where the associated reaction is given by
kib
species k þ species l!species i
By choosing appropriate length, velocity and concentration scales, L, U1 and fpl 1, respectively, such that
u¼
z~
u~
v~
R~
; v¼
; R¼ ; z¼
L
U1
U1
L
fi ¼
f~i
fpl 1
the non-dimensional diffusion coefficient Di is defined as
Di ¼
~i
D
~
Dpl
If only a single bulk reaction is modelled, then the nondimensional production rates for platelets, activated
platelets and thrombin are given as
W pl ¼
W act pl ¼
W th ¼
~ pl L kb L
W
¼
½fpl Š½fth Š ¼ Dm½fpl Š½fth Š
U1
U1
~ act pl L
W
¼ 2Dm½fpl Š½fth Š
U1
~ th L
W
¼0
U1
Dm is the Damkohler number, a ratio of fluid transit time
to chemical reaction time. The diffusion coefficients for
cells moving in blood plasma are extremely low and, thus,
Pe is correspondingly high. For physiological conditions,
the velocity boundary-layer thickness, d , Re 1=2 ; is large
compared to the species mass-transfer boundary layer,
1 , Pe 1=2 as shown in Fig. 2. The velocity profile can,
therefore, be assumed to have a linear form and the
velocity vector field may be evaluated independently.
For the cases presented here, the velocity field is
assumed known a priori and the wall shear stress evaluated
from this. For the particular case of a stagnation point flow,
the non-dimensional wall shear stress function, tw(R ),
may be analytically generated using the Heimenz solution
(Schlichting, 1960). However, this solution is only valid in
the non-dimensionalised form of the species equation is
given as
u
›f i
›fi Di ›2 fi
þv
¼
þ Wi
›R
›z
Pe ›z 2
i ¼ pl; th; act pl ð10Þ
where Pe, is the Peclet number defined as
Pe ¼
U1L
~ pl
D
FIGURE 2 Comparison and characteristic depths of the boundary layer
for momentum, d, and mass transfer, 1.
PLATELET ACTIVATION AND DEPOSITION
99
streamline the shear stress is a linear function of the radial
coordinate, tw ðRÞ ¼ aR; say. Substitute this form into
Eq. (12) then it is easy to show that
lim½bðRފ ¼
R!0
FIGURE 3 Wall shear stress, tw(R ) as a function of non-dimensional
radial co-ordinate for the axi-symmetric stagnation point flow.
1
c ¼ tw ðRÞRez 2
2
ð11Þ
and an appropriate von Mises transformation (Schlichting,
1960) a similarity variable, h, defined by
{tw ðRÞ}1=2 ðPeReÞ1=3
h ¼ zh Ð
i1=3 ¼ zbðRÞ
R
9 0 {tw ðgÞ}1=2 dg
ð12Þ
may be used, where the non-dimensional wall shear stress
is given by
tw ðRÞ ¼
~
t~w ðRÞ
2
rU 1
and the Reynolds number Re is defined as
Re ¼
rU 1 L
m
A full derivation of the similarity variable is given in the
appendix. We note that close to the stagnation point
ð13Þ
So that even though the wall shear stress tends to zero at
the stagnation point, the similarity variable is defined at
R ¼ 0: In comparison with other workers where the flow
in the Cartesian co-ordinate system is of a Poiseuille type,
then tw ðxÞ is of a constant value and the similarity variable
has the form given by
h¼
a small neighbourhood of the stagnation streamline where
the shear stress is a linear function of the radial coordinate, R. It was felt important, especially when
comparing with the results of Affeld et al. (1995), that a
much wider domain should be considered. Hence tw(R )
was derived from a numerical solution to the Navier–
Stokes equations (David et al., 2001b). The computational
domain mimicked the experiment by Affeld et al., (1995),
with no slip boundary conditions on the lower and upper
walls, a specified parabolic profile at the pipe entrance
(positioned at right-angles to the adherent surface) and a
zero gradient condition at the outlet. In the interests of
brevity, we refer to the work of David et al. (2001b) for a
description of the computational domain. Figure 3 shows
the wall shear stress as a function of the radial co-ordinate,
R. This compares very well with that given by Affeld et al.
(1995) as shown by Thomas (2000).
Results for a single species model (David et al., 2001b)
have shown that for high Pe with the stream-function
definition given by
aRePe 1=3
6
RePetw
9ðx 2 x0 Þ
1=3
y
ð14Þ
with tw ðxÞ; the non-dimensional constant wall shears
stress in the stream-wise direction x. Using either Eq. (12)
or (14) to define the similarity variable depending on
whether a cylindrical or Cartesian co-ordinate system is
considered. In both cases, the non-dimensional species
conservation equation becomes for the ith species mass
fraction
Di
d2 f i
df i
þ 3h 2
¼ M½fk Š½f1 Š
2
dh
dh
ð15Þ
Pe
M ¼ Dm 2 :
b ðRÞ
The resulting ordinary differential equations can be used
in a variety of fluid flow cases where tw(R ) is known a
priori and is well behaved. The flux boundary condition at
the surface is of the form (for the cylindrical case)
›fi dfi ›h dfi
¼
¼
bðRÞ ¼ ki fi wall
›z
dh ›z
dh
ð16Þ
ki L
ki ¼
fpl 1 D~ i
Here ki may be thought of as a Sherwood number for the
ith species. However, since only a single reaction occurs
in the bulk fluid, the Damkohler numbers Dmi can
be considered to be equal for all i, Dmi ¼ Dm; ;i ¼ pl; th:
Finally, a boundary condition modelling the species mass
fraction at a large distance away from the surface
is needed. We assume that far away from the
adhesion surface the concentration of species is a constant.
So that
h ! 1; fi ðhÞ ! fi 1
ð17Þ
We note that there are two limiting conditions to the
problem. Firstly, where the Damkohler number is small
and secondly, where it is large. It is normally the case
that the activation of platelets is fast and hence the
Damkohler number is large for values of L=U 1 , Oð1Þ:
100
T. DAVID et al.
TABLE I 0 # h , h* (18)
Platelets
h ¼ 0;
h ¼ h* ;
Activated platelets
›fpl
›h
¼
kwp
bðRÞ fpl
h ¼ 0;
fpl ¼ 0:0
h ¼ h* ;
›fact pl
›h
¼
Thrombin
kwa
bðRÞ fact pl
h ¼ 0;
›fth
›h
wt
¼ 2 bkðRÞ
½kwa fact pl þ kwp fpl Š
fact pl ¼ 1:0
Large Damkohler number cannot occur for negligibly
small U1, since this would invalidate the similarity
variable. We investigate first the case of Dm ! 1
where we obtain an analytical solution and then use a
numerical procedure for the case for intermediate values
of Dm. The analytical and numerical solutions are then
compared.
We are now in a position to solve the resulting equations
(19) subject to the boundary conditions (18) and (21). For
the individual domains the solutions are
0 # h < h*
Platelets
Analytical Solution for Dm ! 1
fpl ¼ 0
For this case, the domain of reaction, where the activation
of platelets by thrombin takes place, is an infinitely small
sheet. Let us assume that the position of this sheet is h*.
The concentration of non-activated platelets at this
position is zero and all platelets have been activated.
The set of ordinary differential equations for platelets,
activated platelets and thrombin become
Activated platelets
fact pl ðhÞ ¼
£
3kwa
3Dact pl bðRÞ þ kwa Gð1=3; h*Þ
ðh
expð2g 3 Þdg
ð22Þ
0
Di
d2 fi
dfi
þ 3h 2
¼ 0 i ¼ 1; . . .3
2
dh
dh
ð19Þ
The RHS of Eq. (19) is zero since all the reaction terms are
now embedded in a infinitesimally thin reaction sheet
and the equation is satisfied in the entire domain excluding
the point h ¼ h*: The domain is split into two sections
0 # h , h* and h* , h , 1: The boundary conditions
are set out in Tables I and II.
To close the system, we have to find the value of h*.
Since the convective velocity for reacting species at the
reaction surface is the same, then thrombin and platelets
can only approach each other by a diffusive process and
only in stoichiometric proportions. This means that the
diffusive mass flux must be in stoichiometric proportions,
thus
dfpl dfth ¼ 2mpl
mth
dh h¼h*
dh h¼h*
ð20Þ
þ
3Dact pl bðRÞ
3Dact pl bðRÞ þ kwa Gð1=3; h*Þ
Thrombin
fth ðhÞ ¼
23kwt
3DAct pl bðRÞ þ kwp Gð1=3; h*Þ
£
! ð23Þ
1=3
g3
Dth Gð1=3; 1Þ
exp 2
dg 2
Dth
3
0
ðh
h* # h < 1
Platelets
fpl ðhÞ ¼
where mth and mpl are the molecular masses of thrombin
and platelets, respectively.
3fpl 1
Gð1=3; 1Þ 2 Gð1=3; h*Þ
ð h
Gð1=3; h*Þ
3
expð2g Þdg 2
£
3
0
ð24Þ
Activated platelets
TABLE II h* # h , 1 (21)
Platelets
Activated platelets
h ¼ h* ; fpl ¼ 0:0
h ¼ h* ; fact pl ¼ 1:0
h ! 1; fpl ! 1:0
h ! 1;
fact pl ! 0:0
Thrombin
h ! 1;
fth ! 0:0
fact pl ðhÞ ¼
3fpl 1
Gð1=3; h*Þ 2 Gð1=3; 1Þ
ð h
Gð1=3; 1Þ
3
expð2g Þdg 2
£
3
0
ð25Þ
PLATELET ACTIVATION AND DEPOSITION
Thrombin
fth ðhÞ ¼
TABLE III
23kwt
3Dact pl bðRÞ þ kwp Gð1=3; h*Þ
£
g3
D Gð1=3; 1Þ
exp 2
dg 2 th
Dth
3
0
ðh
1
3
!
The above equation is solved using a Maplee root finding
algorithm for specific values of the molecular masses and
the surface reaction rates.
Numerical Solution for Intermediate Dm
For the case where the Damkohler number is neither large
nor small, we use a numerical solution method. The
domain is treated as a continuous range and here the set of
differential equations governing the concentration of
species are written as
d2 fpl
dfpl
þ 3h 2
¼ 2Mpl ½fth Š½fpl Š
dh 2
dh
Dact pl
d2 fact pl
dfact pl
þ 3h 2
¼ Mact pl ½fth Š½fpl Š
dh 2
dh
ð28Þ
d2 fth
dfth
þ 3h 2
¼0
Dth
dh 2
dh
Mpl ¼ Mact pl ¼
Kb (m s21)
Pe
kwt
~ th
D
kwp
~ pl
D
kwa
~ act pl
D
5.0 £ 106
1.0 £ 106
0.1
10.0
1.0
1.0
2.0
1.0
ð26Þ
Ðx
Here, Gða; xÞ ¼ 0 e 2t t a21 dt is the incomplete gamma
function. The concentrations are continuous across the
reaction sheet; however, the derivatives are not, except
that for thrombin where all its derivatives are continuous.
This is due to the fact that thrombin is essentially a catalyst
in the reaction. To complete the analysis for large
Damkohler number we use the stoichiometric Eq. (20) for
the fluxes of thrombin and platelets and substitute in the
derivatives of the known solutions (evaluated as h ! h*^ )
and the masses for thrombin and platelets. To find h*, we
require the root of the following equation
mth
2h* 3
1
1
kwa exp
G ; 1 2 G ; h*
mpl
Dth
3
3
ð27Þ
ÿ
1
¼0
2 exp 2h* 3 3bðRÞ þ kwp G ; h*
3
Dpl
101
kb L Pe
U 1 b 2 ðRÞ
The boundary condition for the numerical model are the
same as those used in the analytical solution for h ¼ 0 and
h ! 1: A NAG (Ltd., N., 1999) routine (D02RAF) is
used to solve the coupled set of o.d.e.’s [Eq. (28)] for a
range of values of the reduced Damkohler number M.
D02RAF solves the two-point boundary-value problem
with general boundary conditions for a system of ordinary
differential equations, using a deferred correction
technique and Newton iteration. In this case, a
continuation parameter was used that incremented the
reduced Damkohler number M from zero to the prescribed
value. The routine also uses a mesh refinement technique
that equidistributes an estimate of the truncation error
across the mesh.
M is a function of both the bulk reaction-rate kb and the
free-stream velocity U1. Thus, variation in the source
term for the conservation of species equation can come
from either a variation in bulk reaction-rate or a change in
U1. It is noteworthy to investigate the relationship
between the reduced Damkohler number and that of the
free-stream velocity U1. We can write
h Ð
i2=3
R
9 0 n* dR
kb L
½PeŠ1=3
M¼
U1
n2*
h Ð
i2=3
R
1=3
9
n
dR
0 *
kb L
U1
22=3
¼
/U 1
n2*
DL
U1
ð29Þ
Here n* is a wall slip velocity defined from the wall shear
stress (see Appendix). From the above, it is seen that
variations in the free-stream velocity U 1 may have a
pronounced effect on M and thus a considerable influence
on the activation of platelets.
Reaction rates for the activation of platelets are not
readily available in the literature, however, experimental
evidence suggests that a figure of order 106 s21 (Strong
et al., 1987) is not uncommon since the overall rate for the
inhibition of thrombin by anti-thrombin III is of the same
value. Using the value of 1.7 £ 10213 m2 s21 for the
diffusion coefficient of platelets and, due to its size a value
10 times higher for thrombin, along with L ¼ 0:003 m;
U 1 ¼ 0:01 m s21 ; provides the characteristic Peclet
number and Reynolds number for the flow conditions
similar to those found in Affeld et al. (1995). It should
be noted that Affeld et al.’s experiment had a value of
Re ¼ 2: Adhesion rates from earlier mathematical models
are available, however (Strong et al., 1987), Table III
provides data used in the solution of both the analytical
and numerical models.
RESULTS
Figure 4 shows platelet, activated platelet and thrombin
concentrations for both the analytical and numerical
solutions using the input data given by Table III and a
reduced Damkohler number of M ¼ 5:0 £ 106 : The root
102
T. DAVID et al.
FIGURE 4 Analytical and numerical solutions for platelet, activated
platelet and thrombin concentrations using the data of Table III and
Damkohler number M ¼ 5:0 £ 106 as a function of the similarity
variable h.
finding algorithm solution for Eq. (27) places the reaction
sheet at h ¼ 3 for the cylindrical stagnation point system
(accurate to three significant figures). Non-activated
platelets diffuse toward the reaction sheet from h . 3;
whilst activated platelets diffuse from the reaction sheet
toward the outer boundary. For values of smaller h, the
concentration of activated platelets maintains a maximum
value of 1 and then decreases towards the surface at h ¼ 0.
The numerical solution shows that platelets are activated
in a very small reaction zone, in this case in the
neighbourhood of h ¼ 3; and this position separates the
inner layer where, apart from very small concentrations of
thrombin, only activated platelets exist and the outer layer
where there are platelets and diffusing activated platelets.
The thrombin concentration is small and is unaffected by
the reaction since it is only a catalyst in the activation
process.
FIGURE 5 Concentrations of activated, non-activated platelets and
thrombin versus similarity variable h with M ¼ 5:0 £ 103 using the
numerical solution.
FIGURE 6 Concentrations of activated, non-activated platelets and
thrombin versus similarity variable h with M ¼ 5:0 £ 101 using the
numerical solution.
When the Damkohler number is of intermediate size
only the numerical solution can be used. From the results
of the numerical model and using identical data to that of
Fig. 4, apart from a reduction in kb to allow a smaller
reduced Damkohler number of M ¼ 5:0 £ 103 ; Fig. 5
shows the concentration of all three species as a function
of the similarity variable h. The maximum concentration
of activated platelets is now reduced, only 90% of that
previously. In addition, the reaction zone is considerably
larger, centred at a different position and occurring over
the approximate range of 0:6 , h , 1:6: The concentration of activated platelets at the adhesion surface is
however similar to that for M ¼ 5:0 £ 106 :
Figure 6 again shows platelet and activated platelet
concentrations but this time with a Damkohler number
(again by reducing kb) of M ¼ 5:0 £ 101 : For this case the
activated platelet concentration is a small perturbation to
the non-reactive case. Almost no activated platelets are
generated.
As an indicator of how thrombin production affects
platelet adhesion, Fig. 7 illustrates the variation of the total
platelet flux to the surface (for activated and non-activated
platelets), q_ ; as a function of the rate of thrombin
production at the adhesion surface per unit adhesion
of platelets, kwt. The value of kb is now such that
M ¼ 5.0 £ 106, as in the case presented by Fig. 4. The
total platelet flux to the wall is defined as
dfact pl dfpl þ
ð30Þ
q_ ¼ bðRÞ
dh
dh h¼0
As the thrombin production (per unit concentration of
activated platelet), kwt, increases, then the flux initially
rises rapidly but tends to a constant value as kwt ! 3: This
plateau is possible due to the mass-transfer boundary layer
being completely composed of activated platelets and
hence saturated.
PLATELET ACTIVATION AND DEPOSITION
103
FIGURE 7 Platelet flux to the adhesive surface [Eq. (30)], evaluated at
R ¼ 0 versus thrombin flux rate kwt for M ¼ 5:0 £ 106 :
FIGURE 9 Platelet flux to the adhesive surface [Eq. (28)], evaluated at
R ¼ 0 versus free-stream velocity U1 for kb ¼ 4 £ 105 and 4 £ 107.
It is important to investigate how the bulk reaction-rate
affects the platelet adhesion at the surface. The variation
of total platelet flux, q̇, as a function of bulk reaction-rate,
kb, is shown in Fig. 8. Here the bulk reaction-axis is
logarithmic. In a similar manner to Fig. 7, the flux
increases as the bulk reaction-rate increases until a plateau
is reached for kb . 106 : Thus all variation of the platelet
flux occurs over a range of approximately 100. In this case
the plateau corresponds to the situation modelled by the
analytical solution given by Eqs. (22 –26).
In the light of the variation of Damkohler number with
both kb and U1, it is also necessary to look at the total
platelet flux to the adhesion surface, q_ ; as a function of
characteristic velocity U1 and this is shown in Fig. 9. Two
curves are plotted for values of the bulk reaction-rate
kb ¼ 4 £ 105 and 4 £ 107. For the case of kb ¼ 4 £ 105 ;
the flux increases to a maximum value where U 1 < 0:6: It
then decreases rapidly over the range 0:6 # U 1 # 2:0:
When the bulk reaction-rate is increased to kb ¼ 4 £ 107 ;
then the platelet flux maximum is reached at U 1 < 2:0:
The shape of the curve is similar for both values of the
bulk reaction-rate and it should be noted that for smaller
values of U1, the curves lie on top of each other. For lower
but increasing values of U1, more platelets are being
convected into the mass-transfer boundary layer, however,
as U1 increases beyond a critical value, then the bulk
reaction cannot be sustained and platelet activation
decreases.
Figure 10 shows the calculated flux of platelets to the
surface for the stagnation point flow case as a function of
the stream-wise co-ordinate R. This shows clearly that the
maximum flux occurs at the stagnation point streamline, in
contrast to that found in the experimental data of both
Affeld et al. (1995) and Karino and Goldsmith (1979). In
addition, the platelet flux for a constant shear flow profile
(Poiseuille flow), where the downstream co-ordinate is x,
is also shown for comparison. For both cases, the flux is
dominated by the shape of the function b(R ) or b(x ) [see
Eqs. (12) and (14)].
For initial platelet adhesion, the steady-state solution
may be used to estimate the number of platelets adhered to
the surface at a certain time; this assumption has been used
previously as shown by Strong et al. (1987).
Figure 11 shows the dimensional number of adhered
platelets per unit area, as a function of axial distances as
evaluated by Sorensen et al. (1999a) with parallel plate
FIGURE 8 Platelet flux to the adhesive surface [Eq. (28)], evaluated at
R ¼ 0 versus platelet activation rate kb for M ¼ 5:0 £ 106 :
FIGURE 10 Platelet flux to the adhesive surface [Eq. (28)], versus nondimensional radial co-ordinate for the stagnation point flow and axial
co-ordinate.
104
T. DAVID et al.
FIGURE 11 Number of adhered platelets per unit area, as a function of
axial distance as evaluated by Sorensen et al. (1999a) with parallel-plate
experiments (after 75 s), compared with that calculated by the present
theory.
experiments (after 75 s), compared with that calculated by
the present theory (b(x ) defined by Eq. (14)). By
comparing these two models, it should be noted that the
kinetic mechanism of Sorensen is considerably more
complex, especially in the inhibition of thrombin in the
bulk flow.
For values of the downstream co-ordinate x . 3; the
agreement is excellent. However, for smaller values of the
downstream co-ordinate, the present theory predicts a
smaller flux. Fogelson and Wang (1996) provided a model
of thrombin production diffusing through the canaliculae
of activated platelets. This can be modelled in a simple
fashion by including a non-zero source term (proportional
to the bulk concentration of activated platelets) in the
conservation equation for thrombin [Eq. (28)]. The
coefficient of proportionality is the same as that for
the surface boundary condition for thrombin. Thus the
activated platelets are a source for thrombin both in
the bulk fluid and at the surface. Figure 12 shows the
concentrations for platelets, activated platelets and
thrombin (magnitude £ 100) for this case where the
FIGURE 13 Log –Log plot of platelet flux versus axial distance
compared with that of Turitto and Baumgartner (1975).
same input data has been used as that for Fig. 5. The
activated platelets, in exuding thrombin, have increased
the concentration of activated platelets and have moved
the reaction zone further from the adhesion surface
(compare with Fig. 5). If the production of thrombin per
unit concentration of activated platelets is increased, then
the reaction surface is moved even further away from the
adhesion surface and the maximum concentration of
activated platelets is increased.
On the basis of a power law relationship between shear
rate and platelet diffusion, Turitto and Baumgartner (1975)
plotted platelet adhesion data from parallel perfusion
chamber experiments on log – log axes. In doing so, they
stated that the initial data point value was too low according
to their theory and put this down to its axial position being
“too close to the vessel edge.” Their straight-line fit,
therefore, did not use this data point. Figure 13 compares
the present theory with the experimental data of Turitto and
Baumgartner where their experiment used a simple flatplate perfusion chamber. Due to the possible uncertainty of
the value of the diffusion coefficients, we can write the
equation for platelet flux in the form
qðxÞ ¼
k
1 þ kax 1=3
ð31Þ
and use a least squares Marquardt – Levenberg algorithm to
fit to the experimental data of Turitto and Baumgartner’s
rate of platelet adhesion (Turitto and Baumgartner, 1975)
via the parameters a and k. For this case, the parameter a
takes into account the constant shear stress found in a flatplate perfusion chamber experiments of Turitto and
Baumgartner. Table IV shows values found for a and k.
TABLE IV
Variable
FIGURE 12 Concentrations for platelets, activated platelets and
thrombin (magnitude £ 100) versus the similarity variable h, the input
data is as that for Fig. 5.
k
a
Value
63% conf. interval
46.18
0.00257
^3.74
^0.000911
,8%
,35%
PLATELET ACTIVATION AND DEPOSITION
Figure 13 shows the data of Turitto and Baumgartner
(1975) along with the least squares fit using Eq. (31). In
addition, using the same type of analysis as that of Turitto
and Baumgartner, Fig. 13 also shows a linear least
squares fit of the form logðqÞ ¼ m logðxÞ þ logðcÞ: The
present theory shows a very good agreement with
experiment.
105
activated platelet concentration behaves in a manner
given by a simple convection diffusion equation, not
unlike Eq. (19). Thus the concentration profile has the
form at the surface given by
fact pl ð0Þ ¼
~ act pl bðRÞ
3D
3bðRÞ þ kwa Gð1=3; h*Þ
ð32Þ
[derived from Eq. (22)] and the flux to the surface is
DISCUSSION
The analytical solution for Dm ! 1 shows excellent
agreement with that from the numerical model. In fact, for
values of h , 3; the curves lie on top of one another and
similarly for h . 5: Asymptotic analysis could be used to
investigate the details of the reaction region when Dm is
large, but not infinite. However, this may be counter
productive since the region would certainly be of the order
of the cell diameter for platelets.
The three different cases where the bulk reaction-rate is
reduced, shown above in Figs. 4– 6, indicate that as the
Damkohler number is reduced, the reaction zone both
widens and moves closer to the adhesion surface. Below a
value of kb ¼ 105 the activation of platelets has virtually
ceased and only non-activated platelets are adhering
and present at the adhesion surface and this is a
limiting condition. Whilst for large values of kb the flux
attains a plateau, again a limiting case, whose value is
essentially controlled by the adhesion rate of activated
platelets kwa.
The platelet flux as given by Eq. (30) is a function of the
thrombin flux emanating from the adhered platelets and is
shown in Fig. 7. Here, as kwt decreases and the flux of
thrombin from the surface tends to zero, then the
concentration of activated platelets reduces and the
platelet flux tends to a limiting case. This is where cells
diffusing toward the surface are only non-activated
platelets in a manner similar to that of the platelets
when no bulk reaction occurs. In contrast, as the value of
kwt gets much larger than unity, then the reaction zone
moves farther away from the surface and there exist
essentially two layers. The inner layer, where only
activated platelets exist, and the outer layer, where there
are platelets yet to be activated. The activated platelets
diffuse toward the adhesion surface from a maximum
value of unity and the situation is similar to that found in
Fig. 4. The outer layer consists of non-activated platelets
diffusing toward the reaction zone, where their concentration is negligibly small, and activated platelets
diffusing outward from the reaction zone. The resulting
flux to the adhesion surface comes from the activated
platelets only and tends to a limiting value as kwt ! 1 as
shown later. The same condition arises for the case of the
reaction rate of activation when kb ! 1 as shown in
Fig. 8. The inner layer is populated by activated platelets
alone and the flux is determined almost entirely by the
rate of adhesion at the surface. For the case of high
or even moderate values of bulk reaction-rate, the
q_ ¼
~ act pl bðRÞ
3kwa D
3bðRÞ þ kwa Gð1=3; h*Þ
ð33Þ
We note that Gð1=3; h*Þ is not too dissimilar in value to
Gð1=3; 1Þ: When the surface adhesion rate, kwa, is high
then the flux is given by
limð_qÞ ¼
kwa !1
~ act pl bðRÞ
3D
Gð1=3; h*Þ
ð34Þ
which is a constant for constant values of either R or x
depending on whether the co-ordinate system is
cylindrical or Cartesian, respectively. Equation (33)
shows that for intermediate adhesion kinetics and high
activation rate, then for there to be a significant reduction
in the flux to the surface we must have that
1,
kwa Gð1=3; h*Þ
3bðRÞ
)
Gð1=3; h*Þ
3bðRÞ
3
{tw ðRÞ}1=2 ðPeReÞ1=3
¼
h Ð
i1=3 < kwa
Gð1=3; h*Þ
R
9 0 {tw ðgÞ}1=2 dg
ð35Þ
The relationship between adhesion rate and fluid
dynamics becomes a simple balance between b(R ) and
kwa. The convective and diffusive terms are taken into
account by the similarity coefficient that shows the
fundamental relationship between wall shear stress and
the diffusion of platelets. This will be true for all 2D
flow conditions provided the flow is fully attached or is
in the neighbourhood of an attachment stagnation
point (wall shear stress is well behaved) and is known
a priori.
In contrast to the reduction in Damkohler number due to
a reduction in bulk reaction rate kb, an increase in U1,
although providing a similar decrease in Damkohler
number, can produce a maximum in the platelet flux. For
this maximum to exist, there must be two competing
effects. As the velocity U1 decreases, b(R ) also decreases
(for all R or x if a 2D system is being considered) and
effectively reduces the platelet flux to the adhesion
surface. However, the Damkohler number increases due to
Eq. (28) and effectively strengthens the reaction source
term whilst in contrast to moving the reaction zone away
from the adhesion surface. The activated platelets have to
diffuse further toward the surface and their concentration
at the adhesion surface thereby reduces. As the velocity
106
T. DAVID et al.
U1 increases, the reaction surface moves toward the
adhesion surface, but the Damkohler number decreases
sufficiently to cause a significant decrease in the amount
of activated platelets generated and the total platelet flux is
reduced. As the bulk reaction-rate increases, the position
of maximum platelet flux also increases.
If the characteristic length, L, is maintained constant,
then Fig. 8 could be thought of as showing the relationship
between platelet flux and a characteristic “global” shear
rate given by U 1 =L: For the stagnation point flow regime
this ratio could be thought of as the “strength” of the
stagnation flow. However, it should be noted that this
“shear rate” is not a local one, but characteristic of the
“experiment” as a whole. Experiments by Turitto et al.
(1979), where the rate of platelet adhesion is plotted
against “global” shear rate, also show this maximum
behaviour for both rabbit and human blood. They
stated that the reason for this maximum was, at the
time, unclear. Although, it should be stated that
the experiments were carried out with a sodium citrate
anticoagulant that has the effect of inhibiting activation by
thrombin.
In comparing the work of Sorensen et al. (1999a,b) with
the present theory, we see that the more complex numerical
kinetic mechanism of Sorensen predicts a larger number of
platelets adhering to the upstream surface than that
evaluated by the present model. There could be several
reasons for this. Firstly, the Sorensen simulation is time
dependent and the reactive surface boundary conditions
take account of a reducing area available for adhesion.
Initial studies with a time-dependent algorithm (David
et al., 2001a) show that this is possibly due to axially
dependent scaling factors multiplying the diffusive and
convective terms for the time-dependent equation. A fuller
investigation is left for a further paper. Secondly, the bulkreaction scheme takes into account the inhibition of
thrombin using the work of Griffith (1982a,b). If this were
incorporated into the present scheme, the first-order
reaction rate for heparin and antithrombin III would
increase as the platelets moved downstream.
In the present model, the activation of platelets has been
taken into account using the reaction of thrombin with
platelets where thrombin has emanated from activated and
adhered platelets. In the work by Affeld et al. (1995) using
a axially symmetric stagnation point flow, it was
hypothesised that the thrombin would be convected
downstream and “at some critical shear rate” induce a
higher (maximum) value of platelet flux away from the
stagnation point streamline, which had been observed in
the experiments. The platelet flux to the surface is
evaluated by Eq. (30). Consequently, the variation with
downstream co-ordinate R varies as b(R ). It has already
been shown (David et al., 2001b) that b(R ) is a monotonic
decreasing function for both stagnation point flow and
for constant wall shear stress conditions. By inspection
of Fig. 10, the maximum non-dimensionalised platelet
flux occurs at the stagnation point streamline ðR ¼ 0Þ;
in contradiction to that found by experiments
(Reininger et al., 1993; Affeld et al., 1995; Reininger
et al., 1996). Although it is recognised that RBCs can
augment the diffusion of platelets, Affeld et al.’s
experiments used platelet-rich plasma and hence contained no RBCs. Nor can the flux of thrombin emanating
from the adhesion surface, assumed by Affeld, alone cause
the particular adhesion distribution found. An additional
phenomenon must be present for the variation in R, found
by experiment, to occur. It has been shown (David et al.,
2001b) that an adhesion rate, which, if defined as a
function of shear rate, can provide the required variation,
as suggested by Rajagopalan et al., 1988; Weiss, 1995).
However, the comparison is not so good far downstream
of the maximum wall shear stress. As in the case of
Sorensen et al., this may be due to the time-dependency of
the adhesion process and experiment.
Experiments by Vaishnav et al. (1983) on determining
the erosion stress of endothelium under a stagnation point
flow show a striking resemblance to the platelet-adhesion
experiments of Affeld and may point the way to showing
the relationship between the local shear stress and platelet
adhesion. Platelet-adhesion reduction was also seen in the
neighbourhood of the stagnation point by Karino and
Goldsmith (1979).
For a stagnation point flow condition, the local shear
rate does vary as mentioned earlier and this variation is
taken into account by the similarity coefficient b(R ). This
would also be the case for any 2D flow where the
wall shear stress varies as a function of the stream-wise
co-ordinate. Hence, for stagnation flows such as that
exhibited by reattachment points downstream of stenotic
vessels, which can be characterised with a certain strength,
or any flow field where the wall shear stress varies, platelet
adhesion may not occur at all if the reduced Damkohler
number given by
M¼
kb L Pe
U 1 b 2 ðRÞ
is decreased sufficiently. For the stagnation point flow
case M is a minimum at the stagnation point streamline
and the bulk reaction mechanism is virtually extinguished.
The presented model has been used, with only a
different shear stress substituted into the similarity
variable definition (implicitly defined by the least squares
fit), for comparison with the experiments of Turitto and
Baumgartner (1975). This comparison shows that the
model agrees very well with the experimental data. In fact,
it predicts that the data point left out by Turitto and
Baumgartner is a true representation of the adhesion
mechanism and not a “rogue” point as assumed by Turitto
and Baumgartner.
CONCLUSIONS
A mathematical model is presented for the convection,
diffusion and reaction of platelets and thrombin. The local
PLATELET ACTIVATION AND DEPOSITION
shear rate is incorporated into the representation by the
definition of a similarity variable such that the model can
allow for any a priori known 2D or axi-symmetric “wellbehaved” wall shear stress. The results show that for large
values of the Damkohler number, which can occur for
high platelet activation reaction or low free-stream
velocity, there exist two layers. The inner layer consists
of activated platelets that diffuse toward the adhesion
surface and an outer layer consisting almost entirely of
non-activated platelets. The layers are separated by the
infinitesimally thin reaction zones where the platelets are
activated. In this high Damkohler number case, the flux of
platelets to the wall is dominated by the adhesion rate of
activated platelets.
The mathematical model provides a simple relationship
between this adhesion rate and the parameters of Peclet
and Reynolds number for a range of kinetic conditions,
reaction or diffusion controlled or intermediate.
As the Damkohler number decreases, the reaction zone
broadens and moves toward the surface. If wall shear
stress is assumed to be a function of free-stream velocity,
then the model shows that, in accordance with the
hypothesis set out by Caro et al. (1971), as the free-stream
(characteristic) velocity increases, the bulk reaction
decreases until no activation takes place and adhesion
essentially stops. For this case, the convection of cells
through the reaction site is so fast that activation, and
hence adhesion, has little time to take place. Local areas of
platelet adhesion may therefore occur for “advantageous”
values of the local wall shear stress as this directly affects
the reduced Damkohler number. Corresponding “in vivo
scenarios” would be in the immediate downstream portion
of stenosed vessels where slow recirculating flow exists as
well as reattachment stagnation points. Similar situations
arise in the sinus region of the aortic root when prosthetic
valves are in place, especially if insufficient washout
occurs.
In varying the free-stream velocity, the platelet adhesion
flux to the surface attains a maximum value. The freestream velocity value at which this occurs varies with the
platelet activation reaction rate.
Comparison with the parallel-plate perfusion model of
Sorensen et al. (1999a,b) shows that the present model
predicts a lower platelet flux close to the start of the
reactive surface. However, this is probably due to
the presented model being of a steady-state form.
Earlier work has shown that if time-dependency is taken
into account the convective and diffusive terms have a
stream-wise scaling factor that provides for a higher
platelet adhesion at low values of the stream-wise
co-ordinate.
The presented model representing both adhesion and
the presence of the activating species thrombin emanating
from either the adhered platelets or non-adhered activated
platelets does not provide the required variation in platelet
flux for a stagnation point flow compared to that seen in
experiments. It is probable that another factor, possibly a
shear-rate dependent adhesion reaction-rate, is in play in
107
contrast to the “critical shear rate” theory proposed by
Affeld et al. (1995).
Acknowledgements
We would like to acknowledge the kind support of the
British Council and the Netherlands organisation for
Scientific Research (NWO).
References
Affeld, K., Reininger, A.J., Gadischke, J., Grunert, K., Schmidt, S. and
Thiele, F. (1995) “Fluid mechanics of the stagnation point flow
chamber and its platelet deposition”, Artif. Organs 19(7), 597 –602.
Caro, C.G., Fitz-Gerald, J.M. and Schroter, R.C. (1971) “Atheroma and
arterial wall shear. Observation. Correlation and proposal of a shear
dependent mass transfer mechanism for atherogenesis”, Proc. R. Soc.,
Lond. Ser. B 177, 109–159.
David, T., Thomas, S., Walker, P.G., et al. (2001a) “Models of platelet
deposition in stagnation point flow”, In: Middleton, J., ed, Computer
Methods in Biomechanics and Biomedical Engineering—3 (Gordon
and Breach, Reading), pp 743 –748.
David, T., Thomas, S. and Walker, P.G. (2001b) “Platelet deposition in
stagnation point flow: an analytical and computational simulation”,
Med. Eng. Phys. 23(5), 299–312.
Fogelson, A.L. and Wang, N.-T. (1996) “Platelet dense-granule
centralisation and the persistence of ADP secretion”, Am. J. Physiol.
270(Heart Circ. Physiol., 39), H1131–H1140.
Friedman, L. and Leonard, E. (1971) “Platelet adhesion to artificial
surfaces: consequences of flow, exposure time, blood condition and
surface nature”, Fed. Proc. 30(5), 1641–1648.
Griffith, M.J. (1982a) “The heparin enhanced antithrombin III/thrombin
reaction is saturable with respect to both thrombin and antithrombin
III”, J. Biol. Chem. 257, 13899–13902.
Griffith, M.J. (1982b) “Kinetics of the heparin-enhanced antithrombin
III/thrombin reaction. Evidence for a template model for the
mechanism of action of heparin”, J. Biol. Chem. 257, 7360–7365.
Hubbell, J.A. and McIntire, L.V. (1986) “Platelet active concentration
profiles near growing thrombi. A mathematical consideration”,
Biophys. J. 50(5), 937 –945.
Karino, T. and Goldsmith, H.L. (1979) “Adhesion of human platelets to
collagen on the walls distal to a tubular expansion”, Microvasc. Res.
17, 238–262.
Kratzer, M.A. and Kinder, J. (1986) “Streamline pattern and velocity
components of flow in a model of a branching coronary vesselpossible functional implication for the development of localized
platelet deposition in vitro”, Microvasc. Res. 31(2), 250 –265.
Leonard, E.F., Grabowski, E.F. and Turitto, V.T. (1972) “The role of
convection and diffusion on platelet adhesion and aggregation”,
Ann. N.Y. Acad. Sci. 201, 329–342.
NAG Ltd. (1999) NAG Fortran Library Routine D02RAF.
Petschek, H., Adamis, D. and Kantrowitz, A.R. (1968) “Stagnation flow
thrombus formation”, Trans. Am. Soc. Artif. Internal Organs 14,
256 –260.
Rajagopalan, S., McIntire, L., Hall, E.R. and Wu, K.K. (1988) “The
stimulation of arachidonic acid metabolism in human platelets by
hydrodynamic stresses”, Biochem. Biophys. Acta 958, 108– 115.
Reininger, A.J., Reininger, C.B. and Wurzinger, L.J. (1993) “The
influence of fluid dynamics upon adhesion of ADP-stimulated human
platelets to endothelial cells”, Thromb. Res. 71(3), 245– 249.
Reininger, C.B., Graf, J., Reininger, A.J., Spannagl, M., Steckmeier, B.
and Schweiberer, L. (1996) “Increased platelet and coagulatory
activity indicate ongoing thrombogenesis in peripheral arterial
disease”, Thromb. Res. 82, 523 –532.
Schlichting, H. (1960) Boundary Layer Theory (McGraw-Hill,
New York).
Schoephoerster, R.T., Oynes, F., Nunez, G., Kapadvanjwala, M. and
Dewanjee, M.K. (1993) “Effects of local geometry and fluid
dynamics on regional platelet deposition on artificial surfaces”,
Arterioscler. Thromb. 13(12), 1806–1813.
Sorensen, E.N., Burgreen, G.W., Wagner, W.R. and Antaki, J.F. (1999a)
“Computational simulation of platelet deposition and activation: II.
108
T. DAVID et al.
Results for Poiseuille flow over collagen”, Ann. Biomed. Eng. 27,
449 –458.
Sorensen, E.N., Burgreen, G.W., Wagner, W.R. and Antaki, J.F. (1999b)
“Computational simulation of platelet deposition and activation:
I. Model development and properties”, Ann. Biomed. Eng. 27,
436 –448.
Strong, A.B., Stubley, G.D., Chang, G., Absolom, D.R. (1987)
“Theoretical and experimental analysis of cellular adhesion to
polymer surfaces”, J. Biomed. Mater. Res. 21, 1039– 1055.
Thomas, S.J. (2000) “The role of flow in thrombogenesis” PhD Thesis,
School of Mechanical Engineering, University of Leeds (Leeds, UK).
Tippe, A., Reininger, A., Reininger, C. and Riess, R. (1992) “A method
for quantitative determination of flow induced human platelet
adhesion and aggregation”, Thromb. Res. 67(4), 407 –418.
Turitto, V. and Baumgartner, H. (1975) “Platelet deposition on
subendothelium exposed to flowing blood: mathematical analysis of
physical parameters”, Trans. Am. Soc. Artif. Internal Organs 21,
593 –601.
Turitto, V., Weiss, H.J. and Baumgartner, H. (1979) “Rheological factors
influencing platelet interaction with vessel surfaces”, J. Rheol. 26,
735 –749.
Vaishnav, R.N., et al. (1983) “Determination of the local erosion stress of
the canine endothelium using a jet impingement method”, J. Biomech.
Eng. 105, 77–83.
Weiss, H.J. (1995) “Flow-related platelet deposition on subendothelium”,
Thromb. Haemostasis 74(1), 117 –122.
Wilson, J.M. and Ferguson, J.J. (1999) “Platelet– endothelial interactions
in atherothrombotic disease: therapeutic implications”, Clin. Cardiol.
22, 687–698.
then the non-dimensional velocity components can then
be written as
u ¼ v2* z
rU 1 L
¼ v2* zRe;
m
ðA4Þ
r z~ 2 dðv2* Þ
z 2 dð~v2* Þ
Re
¼2
v¼2
2 m dR~
2 dR
We define two new variables thus,
j ¼ R;
z ¼ v* zRe
ðA5Þ
and substituting the velocity expressions into the
convective part of the conservation equation we have
u
›fpl
›fpl
›fpl Re 2 2 ›2 fpl
þv
¼ v2* zRe
v
¼
›R
›z
›j
Pe * ›z 2
ðA6Þ
z ›fpl Re 2 ›2 fpl
)
¼
v * ›j
Pe ›z 2
By transforming again through the variables
Pe 1=3
s¼z
Re 2
APPENDIX
x¼
TRANSFORMATION OF THE CONSERVATION
INTO THE O.D.E. BY SIMILARITY VARIABLE
ðA7Þ
ðj
v* dj
j0
and
For brevity we look at only the cylindrical case in detail.
The non-dimensionalised conservation equation for the
platelet mass fraction can be written as
u
›fpl
›fpl
1 ›2 fpl
þv
¼
›R
›z
Pe ›z 2
ðA1Þ
The velocity profile is assumed to be of a linear form since
the ratio of mass-transfer boundary-layer thickness to the
viscous boundary-layer thickness is small. A streamfunction c is thus defined as
r v~ 2
c~ðR; zÞ ¼ * z~ 2
2m
ðA2Þ
where v* is the “friction” velocity defined by
v~ * ¼
t~w ðRÞ
r
1=2
s
ð9xÞ1=3
ðA8Þ
Upon substituting into Eq. (A6) we have, with a little
algebra,
d2 fpl
dfpl
þ 3h 2
¼0
2
dh
dh
the similarity variable h can be written now in terms of the
initial variables thus,
{tw ðRÞ}1=2 ðRePeÞ1=3
h ¼ zh Ð
i1=3 ¼ zbðRÞ
R
9 0 {tw ðzÞ}1=2 dz
ðA9Þ
Using a stream-function given by
ðA3Þ
t~w ðRÞ; a wall shear stress function whose values are
known a priori. It should be noted that for the axisymmetric case the stream-function given by Eq. (A2) is
not the normal Stokes stream-function. If t~w ðRÞ is nondimensionalised by rU 21 so that
v2* ¼ tw ðRÞ
h¼
r v2 ð~xÞ
c~ð~x; y~ Þ ¼ * y~ 2 ;
2 m
a similar method provides the similarity variable for the
Cartesian system.